Integers - High School Math

We are subtracting here, so it is important to remember that subtracting is really just adding the same number with opposite sign. Thus we can think of the problem as the following:

, which we know is equal to .

Recall that subtracting integers is equivalent to adding the inverse. The inverse of a negative number is the positive number with the same magnitude.

Thus, our problem is equivalent to .

Recall that subtracting is equivalent to adding the inverse:

The inverse of a negative number is a positive number:

Recall the placement of numbers on the number line. is spaces to the left of , and then we add , resulting in movement to the right.

When we add these spaces we end up going spaces back to and then additional spaces to end at .

Recall that subtracting a negative number is equivalent to adding the opposite. Thus,

Recall that subtracting negative numbers is equivalent to adding the opposite. Then, we have that:

Combine like terms:

= (5 + 7) + (-2x - 4x - 9x)

= 12 + (-15x)

= 12 - 15x

To combine like terms, combine the numbers as you would a normal addition or subtraction problem:

Therefore, = 8x - 8y.

Adding and subtracting negative integers is confusing if you don't use parentheses right. As long as you remember to keep them in the right places and stay neat, it is easy. Subtracting negative numbers is the same as adding a positive number, and adding a negative number is the same as subtracting a positive number. So, we could rewrite the equation like this:

To simplify a problem like the example above we must combine the like termed variables.

Like terms are the numbers that have the same variable, in this example, and .

Separate the s to get

Then perform the necessary addition to get

Then separate the ’s to get

Then perform the necessary subtraction to get

We then combine our answers to have the simplified version of the equation .

Let's start with our equation, .

Subtract from both sides:

Combine our s:

Divide both sides by :

However, the problem is asking for not , so multiply both sides by :

X is the only term that appears twice in the equation. Since both are positive numbers, we can add the X terms to simplify the equation.

Remember, adding a negative number is the same as subtracting a positive number. That means we can think of this problem as .

If we think of this on a number line, we start at , move left three units because of the , then move left again by one more unit for the , giving us a total of .

Here, we are interested in combining like terms. Like terms are those with the same variable,.

When combining terms with in them, we add the coefficients. Thus, we have , and .

Therefore, we have .

When comparing negative numbers, it is important to remember that the numbers with the larger absolute value (greatest numerical term) are actually more negative. Though it seems that would be the largest number in this case, it is actually the smallest, as it is the farthest down the number line in the negative direction. The greatest number is the one with the smallest absolute value, which is .

Remember, also, that zero is greater than any negative number!

Find the least common denominator (LCD) and convert all fractions to the LCD. Then order the numerators from the smallest to the largest

So the correct order from smallest to largest is

An easy way to solve this problem is to define as 1, which is an odd integer. Plugging in 1 into each answer yields as the correct answer, since

4 is an even integer, which is what we are looking for.

When comparing negative numbers, recall the number line. Numbers that are "more negative", or negative with a large absolute value, are really very small. Thus, comparing negative numbers can sometimes seem counterintuitive. In this case, since we are comparing two negative numbers, the number with the larger absolute value is actually the smaller number. Therefore, the greater number is .

Recall that when comparing negative numbers, those numbers that are "more negative" are actually further to the left on the number line and thus are smaller. Therefore, even though , we have that .

Therefore is larger.